The present invention pertains generally to imaging devices and more specifically to coded aperture imaging of non-focusable radiation.
The concept of using a coded aperture to image nonfocusable radiation was first introduced by L. Mertz and N. Young, Proc. Conf. Optical Instruments and Techniques, London 1961, p. 305 and later implemented with random arrays and by R. H. Dicke, Astrophys. J. 153, L 101 (1968) and J. G. Ables, Proc. Astron. Soc. Aust. 4, 172 (1968). In the formulation by Dicke and Ables, the oepning of a single pinhole camera was replaced by many randomly arranged pinholes which are collectively known as the aperture. As shown in FIG. 1, each point of a self-luminous object projects a shadow of the aperture on the recorded image plane (picture). Subsequent correlation processing of the picture yields the reconstructed image which resembles the original self-luminous objects.
There are two primary motivations for using a coded aperture approach as shown in FIG. 1. The original motivation was to obtain an imaging system which maintained the high angular resolution of a small single pinhole but produced images that have a signal-to-noise ratio commensurate with the total open area of the aperture. Since coded aperture imaging is often applied to nonfocusable radiation sources which are weak, a single pinhole camera would require a very large opening in order to obtain a reasonable signal-to-noise ratio. However, such a large hole would preclude desired angular resolution.
By using a plurality (N) of pinholes in the aperture, high transmission characteristics can be obtained on a picture which consists of N overlapping images of the object. For a point source, the coded aperture technique can improve the signal-to-noise ratio by approximately .sqroot.N when compared to the single pin-hole camera. Since N can be as large as 10.sup.5, the goal of an improved signal-to-noise ratio is obtainable.
The second primary motivation for using coded apertures is to perform tomography as disclosed by H. H. Barrett et al., Appl. Opt. 12, 2686 (1973). As disclosed, object points at different distances from the aperture cast shadows of the aperture onto the picture with different over-all sizes. The particular depth of a point in the object can be reconstructed by treating the picture as if it were formed by an aperture scaled to the size of the shadow produced by the depth of the point. This property of coded apertures is particularly beneficial in medical applications, although uses in industrial inspection are apparent. Other obvious uses of the coded aperture imaging technique for both two-dimensional and three-dimensional imaging include imaging of cosmic sources, analyzing the configuration of concealed nuclear material, viewing laser fusion events, e-beam fusion events, medical applications, e.g., viewing injected radioisotopes, etc., viewing reactor cores, or imaging any other source of nonfocusable radiation.
The recorded image, or picture, is usually not recognizable as the object since the many pinholes cause the picture to consist of many overlapping images. In order to be useful, the picture must be subjected to a reconstruction operation which compensates for the effects of the imaging system. The reconstruction procedure is designed to give the location and intensity of each source in the field of view. Basically, this is accomplished by detecting the location and strength of the aperture pattern in the picture. The analysis methods which have been developed can be categorized as either deconvolution methods or correlation methods. Each of these methods has been heuristically developed in the manner disclosed below.
If the recorded picture of FIG. 1 is represented by a function P, the aperture A and the object O, EQU P=(O*A)+N (1)
where * is the correlation operation and N is some noise function. In the deconvolution methods, the object is obtained as follows: EQU O=RF.sup.-1 [F(P)/F(A)]=O+RF.sup.-1 [F(N)/F(A)], (2)
where F, F.sup.-1, and R are, respectively, the Fourier transform, the inverse Fourier transform, and the reflection operator.
A major disadvantage of the deconvolution method is that F(A) often times contains small terms which can cause the noise to dominate the reconstructed object. In fact, it has empirically been determined that roughly 15% of the Fourier transforms of a 32.times.32 random array have at least one term which is zero. Although it is possible to avoid these particular arrays, it appears that it is a general property of large binary random arrays to have some small terms in their Fourier transform. Although the noise has been reduced by using Wiener filtering, disclosed by J. W. Woods et al., IEEE Trans. Nucl. Sci. NS-22, 379 (1975), the major problem of the deconvolution method is that there remains a possibility of small terms in the F(A) resulting in an unacceptably noisy reconstruction.
In the correlation method, the reconstructed object is defined to be EQU O=P*G=RO*(A*G)+N*G (3)
where G is called the postprocessing array and is chosen such that A*G approximates a delta function. Normally G is a binary array and is selected such that A*G has desirable properties, rather than being the convolutional inverse function (A.sup.-1). If A*G is a delta function O=O+N*G, and the object can be perfectly reconstructed except for the presence of the noise term. Note that the noise term in the above equation does not have singularities as in the deconvolution method, which is an advantage of the correlation method over the deconvolution method.
The expectation of obtaining a roughly .sqroot.N improvement in the signal-to-noise ratio has not, however, been realized since A*G in general is not a delta function. When this is true, a point on the object will contribute A*G to the reconstructed object instead of a delta function. Thus, even when no background noise is present, the signal-to-noise ratio for a point source becomes a fixed number which is the ratio of the central peak in A*G to the noise in A*G, i.e., the square root of the variance of the sidelobes which is referred to as artifacts. These artifacts, in this manner, limit the possible signal-to-noise ratio improvement.
The obtainable signal-to-noise ratio is even further decreased when the point source is changed to an extended source. For an extended source, the artifacts from all points in the object contribute noise to each point in the reconstructed object. The result is a low signal-to-noise ratio which cannot be improved since the noise is set by the structure in A*G rather than by counting statistics or background levels. In fact, the obtainable signal-to-noise ratio for the coded aperture technique can often times be smaller than the obtainable signal-to-noise ratio for a comparative single pinhole camera for extended sources.
Nonredundant arrays have the property that their autocorrelations, i.e., A*A, consist of a central spike with the sidelobes equal to unity to some particular lag L, and either zero or unity beyond that point, as disclosed by M. J. E. Golay, J. Opt. Soc. Am. 61, 272 (1970). Although this autocorrelation technique used with nonredundant arrays approaches a true delta function, a true delta function has all sidelobes to infinite lags equal to zero. The fact that the sidelobes are equal to a constant value such as unity, affects the reconstructed object by the addition of a removable dc level. By removing the dc level, a true delta function is obtained. However, the sidelobes of the nonredundant array do not have a constant value and hence the reconstructed object necessarily contains artifacts.
The property of the nonredundant array which allows the sidelobes to be flat is apparent by measuring the separation between possible pairs of holes. Each separation (to L) occurs only once, rendering the separations nonredundant. This property, however, limits the number of holes which can be placed in the array. For example, a large nonredundant array having 24 holes has an open area density of only 0.03. This lack of holes in the aperture does not function to significantly increase the transmission characteristics of the aperture over a single pinhole.